Question

Suppose $lo{g}_{a}b+lo{g}_{b}a=c$. The smallest possible integer value of $c$ for all $a,b>1$ is

• A. 4
• B. 3
• C. 2
• D. 1

Answer 1

Answer: (C)

We have,
$lo{g}_{a}b+lo{g}_{b}a=c$
$AM\ge GM$
$\therefore \frac{lo{g}_{a}b+lo{g}_{b}a}{2}\ge \sqrt{lo{g}_{a}blo{g}_{b}a}$
$\Rightarrow \frac{c}{2}\ge \sqrt{\frac{logb}{loga}\times \frac{loga}{logb}}$
$\Rightarrow \frac{c}{2}\ge 1$
$\Rightarrow c\ge 2$
$\therefore$ Smallest positive integer value of $c=2$.